Convergence of fmincon using Interior Point Method

My doubt is both mathematical and related directly to Matlab. I've been studying the conditions needed in general for the Interior Point Methods to converge to a minimum, and they are tipycally very simple (to have objective and restricion functions continously differentiable). But this is true depending on how you solve each step, and else it is required for the minimum proposed by the algorithm to satisfy the linear independence constraint qualification. I can't find if Matlab checks this condition. So my question is about that last point, but more in general it's about under which conditions you can be sure that the optimum found by Matlab it is really an optimum.

Thanks in advance, if something is misunderstood I can try to explain it better (my english is not so complete).